How to create a Round Edges Effect on an object?
If you create a new default UV map for an object, each quad face is given the entire UV canvas.

Disclaimer here, I think this only happens if you have first deleted every other UV map on the object, otherwise the previous UV map just gets duplicated to the new slot.  So if you are confident that you can unwrap your object again, it should be easy at this point to find or create a normal map that will give us this effect.  I just grabbed one from image search results for educational purposes.
You can create a second UV map to do a full unwrap if needed, so for argument's sake I'm using the UV Map node to draw from a specific UV map for this.


Hopefully this works for the Unity workflow, or at least gives you some ideas.